Number Theory Seminar

Tate, in his 1950 doctoral thesis, recast the theory (Euler 1740, Riemann 1850) of the Riemann Zeta function, and more generally the theory of Hecke L-functions (Hecke 1915), in terms of Fourier Theory on the line.

An influential 1964 paper due to Weil and many works of Howe starting in the 1970's, tell us that the symmetries (including Fourier transform) of functions on the line are given by an object called the oscillator (Weil) representation of the Jacobi group (semi direct product of Symplectic group and Heisenberg group).

In this talk, we do the logical step, and interpret Tate's thesis using the oscillator representation. Various gadgets appearing in Tate's thesis become spaces of equivariant functionals on the oscillator representation w.r.t. different subgroups of the Jacobi group.